Problem: Simplify and expand the following expression: $ \dfrac{3}{5r - 10}+ \dfrac{2}{r + 1}+ \dfrac{4}{r^2 - r - 2} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{3}{5r - 10} = \dfrac{3}{5(r - 2)}$ We can factor the quadratic in the third term: $ \dfrac{4}{r^2 - r - 2} = \dfrac{4}{(r - 2)(r + 1)}$ Now we have: $ \dfrac{3}{5(r - 2)}+ \dfrac{2}{r + 1}+ \dfrac{4}{(r - 2)(r + 1)} $ The least common multiple of the denominators is: $ 5(r - 2)(r + 1)$ In order to get the first term over $5(r - 2)(r + 1)$ , multiply by $\dfrac{r + 1}{r + 1}$ $ \dfrac{3}{5(r - 2)} \times \dfrac{r + 1}{r + 1} = \dfrac{3(r + 1)}{5(r - 2)(r + 1)} $ In order to get the second term over $5(r - 2)(r + 1)$ , multiply by $\dfrac{5(r - 2)}{5(r - 2)}$ $ \dfrac{2}{r + 1} \times \dfrac{5(r - 2)}{5(r - 2)} = \dfrac{10(r - 2)}{5(r - 2)(r + 1)} $ In order to get the third term over $5(r - 2)(r + 1)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{4}{(r - 2)(r + 1)} \times \dfrac{5}{5} = \dfrac{20}{5(r - 2)(r + 1)} $ Now we have: $ \dfrac{3(r + 1)}{5(r - 2)(r + 1)} + \dfrac{10(r - 2)}{5(r - 2)(r + 1)} + \dfrac{20}{5(r - 2)(r + 1)} $ $ = \dfrac{ 3(r + 1) + 10(r - 2) + 20} {5(r - 2)(r + 1)} $ Expand: $ = \dfrac{3r + 3 + 10r - 20 + 20}{5r^2 - 5r - 10} $ $ = \dfrac{13r + 3}{5r^2 - 5r - 10}$